Concerning the premises of your second rule of inference:
- $a\qquad\qquad\qquad$ Premise
- $b \to \lnot a\qquad$ Premise
- $\lnot\lnot a\qquad\qquad$ (double negation)
- $\therefore \lnot b\qquad\qquad$ by Modus Tollens using 2. and 3.
As you can see, this is merely an application of Modus Tollens. Given an implication, say "A", the negation of the consequent of "A" implies the negation of the antecedent of "A".
Note: we can also take 1. and 2. to be assumptions, and 4. the conclusion derived from those assumptions, from which we can establish the implication (rule of inference):
$$[a \land (b \to \lnot a)] \implies \lnot b.$$
Perhaps your preference for this rule of inference is due to your familiarity with Modus Tollens?
But note that in this rule of inference, for your "premises", you have the conjunction of an implication ($b \to \lnot a$) with the assertion that $a$, where $a$ is not tied up in an implication. It "stands alone."
On the other hand, the second rule of inference involves the conjunction of two implications for premises; you do not have assertion a to use as a premise, nor b, in a "stand alone" form. Formally, we can establish this rule of inference as follows:
$[(b \rightarrow a) \land (b \rightarrow \lnot a)] \implies \lnot b$
- $ \qquad | \underline{[(b \rightarrow a) \land (b \rightarrow \lnot a)]}\qquad $ Assumption/Premise
- $\qquad | b \rightarrow a\qquad $ 1. Conjunction Elimination
- $\qquad | b \rightarrow \land \lnot a \qquad $ 1. Conjunction Elimination
- $\qquad \qquad |\underline{b\qquad} $ Assumption
- $\qquad \qquad |\ a \qquad$ 2, 4, Modus Ponens
- $\qquad \qquad | \lnot a\qquad$ 3, 4, Modus Ponens
- $\qquad\qquad |\underline{ a \land \lnot a} \qquad$ 5, 6, Conjunction Introduction
- $\qquad \|\underline{\therefore \lnot b \qquad}$ 4 - 7, Negation Introduction (contradiction)
- $ [(b \rightarrow a) \land (b \rightarrow \lnot a)] \implies \lnot b \qquad$, 1 - 8 Conditional Introduction
"Negation Introduction": (8) Based on the fact that our assumption "$b$" led to a contradiction (specifically, $(a \land \lnot a))$, it follows that our assumption "$b$" must thereby be false. So $\lnot b$ must be true, and we are justifying in asserting ("introducing") the negation of "$b$": $\lnot b$.
You have likely gone through such proofs before; I thought by working with each of the rules of inference, perhaps you'd notice differences, as well as similarities.
The two argument forms (rules of inference) you present each have their uses, and it is good to understand the logic behind each of them. I can understand that you see similarities, but there are keen (though subtle) differences in the ways in which each can be used, or when the use of one over the other is most fitting. This often depends on the information (premises) you have available to use in a proof, and how "cumbersome" the use of one way be when compared to the other. By being familiar with all the "tools" available (i.e., understanding equivalences, all the rules of inferences), you're in a much better position to recognize when one is most applicable.
ADD IN
I claimed, in my comment to the answer provided by the OP, that one can adapt the user's preferred rule of inference to the "usual" rule (usual as judged by user). I demonstrate this below.
- $|\;a\qquad$ Premise
- $|\;\underline{b \Rightarrow \lnot a}\qquad$ Premise
- $\quad | \underline{b\qquad\qquad}$ Assumption
- $\quad | a\qquad\qquad$ 1. Replication
- $\quad | \lnot a \qquad$ 2, 3 Modus Ponens
- $\quad |\underline{a \land \lnot a}\qquad $4,5 Conjunction Introduction
- $|\; \lnot b \qquad\qquad$ 3-6 Negation Introduction (contradiction inference)